Scientific theories

A scientific theory can be defined as a set of logical propositions that posits causal relationships between observable phenomena.

Scientific theories

The concept of a scientific theory is not a unitary concept though. As an example, Meehl (1986) lists three kinds of theories:

• Functional-dynamic theories which relate “states to states or events to events”. For instance, we say that when one variable changes, certain other variables change in such and such ways.
• Structural-compositional theories in which the main idea is to explain what something is composed of, or what kind of parts it has, and how they are put together.
• Evolutionary theories which are about the history and/or development of things (e.g., Darwin’s theory, Wegener’s theory of continental drift, the fall of Rome, etc).

First problem: We can not confirm theories

Second problem: We can not (strictly) falsify theories

We can not confirm theories, but maybe we can at least think of a way of disproving them? According to Popper’s view, a theory can be considered as falsifiable if it can be shown to be false. But what does it mean for a theory to be false?

1) Theoretical and statistical models

A statistical model is a device that connect theories to data. It can be defined as an instantiation of a theory as a set of probabilistic statements (Rouder et al., 2016).

1) Theoretical and statistical inference

Causal and inferential relations between substantive theory, statistical hypothesis, and observational data (figure from Meehl, 1990).

2) Measurement error

The logic of falsification is pretty simple and rests on the power of the modus tollens. This argument (whose exposition, for some reason, usually involves swans) can be presented as follows:

• If my theory \(T\) is right, then I should observe these data \(D\).
• I observe data that are not those I predicted \(\neg D\).
• Therefore, my theory is wrong \(\neg T\).

2) Measurement error

According to Einstein, neutrinos can not travel faster than the speed of light. Thus, any observation of faster-than-light neutrinos would act as a strong falsifier of Einstein’s special relativity. In 2011 however, a large team of respected physicists announced the detection of faster-than-light neutrinos (cf. the Wikipedia article).

3) Probabilistic hypotheses

Another problem arises from a misapplication of deductive syllogistic reasoning (a misapplication of the modus tollens). The problem (the “permanent illusion,” as put by Gigerenzer, 1993) is that most scientific hypotheses are not really of the kind “all swans are white” but rather of the form:

• Ninety percent of swans are white.
• If my hypothesis is correct, we should probably not observe a black swan.

3) Probabilistic hypotheses

Consider the following argument (Cohen, 1994; Pollard & Richardson, 1987):

• If a person is an American, he is probably not a member of Congress.
• This person is a member of Congress.
• Therefore, he is probably not an American.

4) The underdetermination problem

Again another problem is known as the Duhem–Quine thesis/problem (aka the underdetermination problem). In practice, when a substantive theory \(T\) happens to be tested, some hidden assumptions, such as auxiliary theories about the instruments we use, are also put under examination (Meehl, 1990, 1978, 1997).

4) The underdetermination problem

Thus, the logical structure of an empirical test of a theory \(T\) can be described as the following conceptual formula (Meehl, 1990, 1978, 1997):

\[(T \land A_{t} \land C_{p} \land A_{i} \land C_{n}) \to (O_{1} \supset O_{2})\]

where the \(\land\) are conjunctions (“and”), the arrow \(\to\) denotes deduction (“follows that …”), and the horseshoe \(\supset\) is the material conditional (“If \(O_{1}\), Then \(O_{2}\)”). \(A_{t}\) is a conjunction of auxiliary theories, \(C_{p}\) is a “ceteribus paribus” clause (i.e., we assume there is no other factor exerting an appreciable influence that could obfuscate the main effect of interest), \(A_{i}\) is an auxiliary theory regarding instruments, and \(C_{n}\) is a statement about experimentally realised conditions (i.e., we assume that there is no systematic error/noise in the experimental settings).

4) The underdetermination problem

\[(T \land A_{t} \land C_{p} \land A_{i} \land C_{n}) \to (O_{1} \supset O_{2})\]

However, although the modus tollens is a valid figure of the implicative syllogism for logical statements (e.g., “all swans are black”), the neatness of Popper’s classic falsifiability concept is fuzzed up by the acknowledgement of the actual form of an empirical test. Obtaining falsificative evidence during an empirical test does not only falsify the substantive theory \(T\), but it does falsify all the left-side of the above statement. In other words, what we have achieved by our laboratory or correlational “falsification” is a falsification of the combined claims \(T \land A_{t} \land C_{p} \land A_{i} \land C_{n}\), which is probably not what we had in mind when we did the experiment (Meehl, 1990).

Consequences

Falsification in science is almost always consensual, not logical (McElreath, 2020). A theoretical claim is considered to be falsified only when multiple lines of converging evidence have been obtained, by independent teams of researchers, and usually after several years or decades of critical discussion. The “falsification of a theory” then appears as a social result, issued from the community of scientists, and (almost) never as a deductive falsification.

Null Hypothesis Significance Testing (NHST)

Let’s say we are interested in height differences between women and men…

set.seed(19) # to get reproducible results
men <- rnorm(n = 100, mean = 175, sd = 10) # 100 men heights
women <- rnorm(n = 100, mean = 170, sd = 10) # 100 women heights

t.test(x = men, y = women)

    Welch Two Sample t-test

data:  men and women
t = 2.4969, df = 197.98, p-value = 0.01335
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.7658541 6.5209105
sample estimates:
mean of x mean of y 
 175.1258  171.4825 

Null Hypothesis Significance Testing (NHST)

We are going to simulate t-values computed on samples generated under the assumption of no difference between women and men (the null hypothesis \(\mathcal{H}_{0}\)).

nsims <- 1e4 # number of simulations
t <- rep(x = NA, times = nsims) # initialising an empty vector

for (i in 1:nsims) {
    
    men2 <- rnorm(n = 100, mean = 170, sd = 10)
    women2 <- rnorm(n = 100, mean = 170, sd = 10)
    t[i] <- t.test(x = men2, y = women2)$statistic
    
}

t <- replicate(n = nsims, expr = t.test(x = rnorm(100, 170, 10), y = rnorm(100, 170, 10) )$statistic)

Null Hypothesis Significance Testing (NHST)

data.frame(t = t) %>%
    ggplot(aes(x = t) ) +
    geom_histogram()

Null Hypothesis Significance Testing (NHST)

data.frame(t = c(-5, 5) ) %>%
    ggplot(aes(x = t) ) +
    stat_function(fun = dt, args = list(df = t.test(men, women)$parameter), size = 1.5) +
    ylab("Probability density")

Null Hypothesis Significance Testing (NHST)

alpha <- .05 # significance threshold (alpha)
abs(qt(alpha / 2, df = t.test(x = men, y = women)$parameter) ) # two-sided critical t-value

[1] 1.972019

Null Hypothesis Significance Testing (NHST)

tobs <- t.test(x = men, y = women)$statistic # observed t-value
tobs %>% as.numeric

[1] 2.496871

P-values

A p-value is simply a tail area (an integral) computed from the distribution of test statistics under (given) the null hypothesis. It gives the probability of observing the data we observed or more extreme data, given that the null hypothesis is true (Wagenmakers, 2007).

\[p[\mathbf{t}(\mathbf{x}^{\text{rep}} ; \mathcal{H}_{0}) \geq t(x)]\]

t.test(x = men, y = women)$p.value

[1] 0.01334509

tvalue <- abs(t.test(x = men, y = women)$statistic)
df <- t.test(x = men, y = women)$parameter
2 * integrate(f = dt, lower = tvalue, upper = Inf, df = df)$value

[1] 0.01334509

Fisher versus Neyman & Pearson

Logic, frequentism, and probabilistic reasoning

The modus tollens is one of the strongest rule of inference in logic. It works perfectly well in science when we deal with hypotheses of the following form: “If \(\mathcal{H}_{0}\) is true, then we should not observe \(x\). We observed \(x\). Then, \(\mathcal{H}_{0}\) is false”.

Interpreting confidence intervals

Confidence intervals are basically regions of significance. Thus, they have to be interpreted as cautiously as p-values, and are submitted to the same flaws.

Preliminary summary (statistics is hard)

Frequentist statistics (e.g., p-values and confidence intervals) make sense under the frequentist interpretation of probability: they refer to long-run frequencies.

Bayes factors

Instead of testing only one hypothesis (the null hypothesis), Bayes factors allow comparing two hypotheses. For instance, let’s say we are comparing two models:

• \(\mathcal{H}_{0}: \mu_{1} = \mu_{2} \rightarrow \delta = 0\)
• \(\mathcal{H}_{1}: \mu_{1} \neq \mu_{2} \rightarrow \delta \neq 0\)

Bayes factors are the new p-values…

Be careful not to interpret Bayes factors as posterior odds… Bayes factors indicate how much we should update our prior odds, in the light of new incoming data. They do not tell us what is the most probable hypothesis, given the data (unless the prior odds are 1:1).

Corroborating impossible claims

In 2011, the prestigious Journal of Personality and Social Psychology published this paper written by Daryl Bem, reporting the results of nine experiments, showing evidence in favour of the existence of precognition in humans.

Questionable research practises

Undisclosed flexibility in data collection, analysis, and interpretation dramatically increases the false positive rates (see also the “garden of forking paths” from Gelman & Loken, 2013).

Scientific cycle and questionable research practises

P-hacking

Pressure to produce (e.g., publish), together with widespread misunderstanding of basic concepts in (philosophy of) statistics, have practical/dramatic consequences on the published literature (Figure from Data colada).

Replicability (and reproducibility) issues

This, amongst other things, prompted an era of (healthy) methodological scepticism, which resulted in a through re-evaluation of classical findings from Psychology and experimental sciences overall. The average replication rate in Psychology was estimated to be around 30%.

Registered reports: Reducing publication bias

Publication bias is defined as the selective publication of findings based on the obtained p-value (or another statistical index). How can we avoid this biased reporting? A simple idea is not to base the accept/reject decision on the statistical significance of the results, but rather on the theoretical relevance and methodological rigour of the study (figure from https://www.cos.io/initiatives/registered-reports).

Registered reports: Reducing publication bias

Do not say “statistically significant”

In 2019, The American Statistician published a special issue on “Moving to a World Beyond p<.05”, with the intention to provide new recommendations for users of statistics (e.g., researchers, policy makers, journalists). This issue comprises 43 original papers aiming to provide new guidelines and practical alternatives to the “mindless” use of statistics. In the accompanying editorial, Wasserstein et al. (2019) provide a first practical recommendation.

ATOM guidelines

Then, they summarise their practical recommendations in the form of the ATOM guidelines:

• Accept uncertainty: we must “countenance uncertainty in all statistical conclusions, seeking ways to quantify, visualize, and interpret the potential for error” (Calin-Jageman & Cumming, 2019).
• Be Thoughtful: we clearly distinguish between confirmatory (preregistered) and exploratory (non-preregistered) statistical analyses. We routinely evaluate the “validity” of the statistical model and we are suspicious of statistical defaults.
• Be Open: we try to be exhaustive in the way we report our analyses and we beware of short-cuts than could hinder important information to the reader.
• Be Modest: we recognise that there is no unique “true statistical model” and we discuss the limitations of our analyses and conclusions. We also recognise that scientific inference is much broader than statistical inference and we try not to conclude anything from a single study without the warranted uncertainty.

Common statistical tests are model comparisons

Model comparison and out-of-sample predictive accuracy

Model comparison and out-of-sample predictive accuracy

Hirotugu Akaike noticed that the negative log-likelihood of a model + 2 times its number of parameters was approximately equal to the out-of-sample deviance of a model…

\[\text{AIC} = \underbrace{-2\log(\mathcal{L}(\hat{\theta}|\text{data}))}_\text{in-sample deviance} + 2K\]

Philosophical oecumenism: Statistical toolbox

Different statistical tools rest on different philosophical frameworks and aim to answer different questions.

Towards a principled workflow: Statistical rethinking

Applying this to empirical data in cognitive sciences

Applying this to empirical data in cognitive sciences

A brief summary

Sensible data analysis is not a linear process. A typical workflow involves the following steps:

• Think hard about a (or several) plausible data-generating process(es).
• Think hard about the available prior knowledge and encode it into your model(s).
• Validate these assumptions using simulation (e.g., prior predictive checking).
• If deemed appropriate, fit the model(s) and update prior knowledge using the Bayesian machinery.
• Assess the validity of the model(s) using simulation (e.g., posterior predictive checking).
• Compare various interesting and competing models.
• Make inference about interesting quantities (using as many statistical indexes as needed).
• This is not a linear process, feedback loops are often needed between these steps.

Further resources

The special issue on “Statistical Inference in the 21st Century: A World Beyond p < 0.05”: https://www.tandfonline.com/toc/utas20/73/sup1.

Everything is fucked: The syllabus, https://hardsci.wordpress.com/2016/08/11/everything-is-fucked-the-syllabus/.

Some examples of ATOMised reporting of statistical modelling (from my own work): https://pubs.asha.org/doi/abs/10.1044/2018_JSLHR-S-18-0006, https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0233282, https://journals.sagepub.com/doi/abs/10.1177/0956797619900336.

Introduction to the Meehlian Corroboration-Verisimilitude theory of science: https://www.barelysignificant.com/post/corroboration1/ and https://www.barelysignificant.com/post/corroboration2/.

The materials of my doctoral course on Bayesian statistical modelling (in French): https://www.barelysignificant.com/IMSB2022/.

Take-home messages